Change of Form of Long Waves. 425 



and therefore 



\*h_ 16K 2 A lJ M2KO 

 Z 3 ~ 3(A + &) 3 



This is a small fraction only when M, the modulus, is small, 

 but the cnoidal waves then resemble sinusoidal waves ; and it 

 is obvious that in this case the equation of their surface may 

 be developed in a rapidly convergent Fourier-series, of which 

 Sir Gr. Stokes has given the first two terms. 



After some more discussion about these cnoidal waves, con- 

 cerning their velocity of propagation and the motion of the 

 particles of fluid below their surface, we proceed to a closer 

 examination of the deformation of long waves. To this effect 



we apply the equation for =— to various types of non-stationary 



waves; and it will appear that, though sinusoidal waves be- 

 come steeper in front when advancing, other types of waves 

 may behave otherwise. 



I. The Formula for -=- . 



In our investigations (in accordance with the method used 

 by Lord Rayleigh, Phil. Mag. 1876, vol. i. p. 257, whose 

 paper has been of great influence on our researches), we start 

 from the supposition that the horizontal and vertical u and v 

 of the fluid may be expressed by rapidly convergent series of 

 the form 



u =f+yfi+ff2+->- 



^=3/01+^2+-.. 



where y represents the height of a particle above the bottom 

 of the canal, and where /,/i, . . . <£i, <j> 2 , . . . are functions of x 

 and t. Of course the validity of this assumption must be 

 proved later on by the fact that series of this description can 

 be found satisfying all the conditions of the problem. 



From one of these conditions, viz., the incompressibility of 



the liquid, which is expressed by ^- + ^- =0, we may 

 deduce 



. __ iy»-i 



Vn n 3a 



and from another, viz., the absence of rotation in the fluid, 



